1 Options and Corporate Finance Options: The Basics Fundamentals of Option Valuation Valuing a Call Option Employee Stock Options Equity as a Call Option on the Firm’s Assets Options and Capital Budgeting Options and Corporate Securities

2 What is an option? A contract that gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time. Most important characteristic of an option: It does not obligate its owner to take action. It merely gives the owner the right to buy or sell an asset.

3 Option terminology Call option – an option to buy a specified number of shares of a security within some future period. Put option – an option to sell a specified number of shares of a security within some future period. Exercise (or strike) price – the price stated in the option contract at which the security can be bought or sold. Option price (also called option premium)– the market price of the option contract.

4 Option terminology Expiration date – the date the option matures. Exercise value – the value of an option if it were exercised today (Current stock price - Strike price). Covered option – an option written against stock held in an investor’s portfolio. Naked (uncovered) option – an option written without the stock to back it up.

5 Option terminology In-the-money call – a call option whose exercise price is less than the current price of the underlying stock. Out-of-the-money call – a call option whose exercise price exceeds the current stock price. LEAPS: Long-term Equity AnticiPation Securities are similar to conventional options except that they are long-term options with maturities of up to 2 1/2 years.

6 Option example A call option with an exercise price of $25, has the following values at these prices: Stock priceCall option price $25 $

7 Determining option exercise value and time value premium Stock StrikeExercise OptionTime value price price value pricepremium $25.00$25.00$0.00 $3.00 $

8 How does the time value premium change as the stock price increases? The premium of the option price over the exercise value declines as the stock price increases. This is due to the declining degree of leverage provided by options as the underlying stock price increases, and the greater loss potential of options at higher option prices.

9 Call premium diagram Stock Price Option value Market price Exercise value

10 Call Option Bounds Upper bound Call price must be less than or equal to the stock price Lower bound Call price must be greater than or equal to the stock price minus the exercise price or zero, whichever is greater If either of these bounds are violated, there is an arbitrage opportunity

11 Upper and Lower bounds

12 Binomial Option Pricing Model Option prices (premiums) can be calculated by using the Binomial Option Pricing Model. It assumes that: The options are European options No dividend payment during the life of the option. BOPM further assumes that in each period there are two possible percentage changes in the stock price.

13 Call option Example BOPM for call options Suppose we have a European call option with one year to expiration and an exercise price of $110. The current market price of the stock is equal to $100. In the course of next year, two rates of return on the stock are possible, -10% and 20%. Thus, the price of the stock can either fall to $90 or rise to $120. The stock is not expected to pay any dividends. The risk-free rate of interest available in the bond market is 10% continuous compounding (EAR=10.52%).

14 Call option Example Security Payoff in Up StatePayoff in Down StateCurrent Price Stock $120.00$ 90.00$ Bond $ $ Call Option $ 10.00$ 0.00?

15 Hedge Portfolio Method Note that in the up state, both stock and call are more valuable. Combine call and stock into a risk-free portfolio. In other words, the payoff from the portfolio will be the same in both states. Write one call option and buy n s shares of stock.

16 Hedge Portfolio Method

17 Hedge Portfolio Method So if we write one call option and buy 1/3 shares of stock, the portfolio value in both states will be $30. This is the same payoff as investing $30/1.1052=$27.15 in bonds. So writing one call option and buying 1/3 shares of stock is equivalent to investing =$27.15 in bonds.

18 Hedge Portfolio Method These two investments should cost same amount of money now, since they have identical payoffs in one year. -Pc + (1/3) 100 =$27.15 Pc = $6.18

19 Risk-Neutral Investor Approach In deriving the call option price one can also use the “risk neutral investor perspective” which reaches the same result as “the hedge portfolio method”. To understand the difference between risk-averse and risk neutral investors consider an uncertain amount to be received at the end of the period.

20 Risk-Neutral Investor Approach Denote the expected value of this uncertain amount by EV. For a risk neutral investor the only important characteristic is the expected value. So to this investor the present value of this future amount is i.e. this investor will discount the expected value by using the risk-free rate.

21 Risk-Neutral Investor Approach On the other hand, the risk averse investor will find the present value by discounting the expected value by R f + Risk Premium i.e. he/she uses a higher discount rate. The Risk Premium depends both on the riskiness of the future cash flow and the risk aversion of the investor. So the risk-averse investor cares about both the expected value and risk.

22 Recall our example Note that buying 1/3 shares of stock and borrowing $27.15 at the risk free rate is equivalent to buying one call option Solve these equations 3 unknowns, n s, B and P c

23 Recall our example Solve these equations, you get

24 Consider the expression for P c Consider the equation for Pc

25 Consider the expression for P c p(u) and p(d)=1-p(u) are called risk neutral probabilities

26 Find expected stock price using risk-neutral probabilities Using risk neutral probabilities, expected return on the stock is the risk free rate

27 Expected return on the stock Let’s assume the true up movement probability is What is the expected return on the stock? E(r s )=12.50%

28 Expected return on the call option What is the expected return on the call option? E(r c )=21.36% Hence the call option is always riskier than the stock. Why? It is a levered investment in the stock. Recall that buying 1/3 shares of stock and borrowing $27.15 at the risk free rate is equivalent to buying one call option

29 risk aversion We do not consider risk aversion It is already included in the story through the current stock price If investors get more risk averse, E(r s ) increases, P s falls, up and down movement rates, risk neutral probabilities and call price all change.

30 Risk-Neutral Investor Approach The procedure is: Use the stock price information to calculate the probability of two possible states, up and down, at each node of the tree as if a risk-neutral investor determined the stock price. Then using the option value at maturity and the calculated risk neutral probabilities, find the call option price simply by discounting the expected value at the risk-free rate.

31 Risk-Neutral Investor Approach

32 Risk-Neutral Investor Approach

33 What Determines Option Values? Stock price As the stock price increases, the call price increases and the put price decreases Exercise price As the exercise price increases, the call price decreases and the put price increases Time to expiration Generally, as the time to expiration increases both the call and the put prices increase Risk-free rate As the risk-free rate increases, the call price increases and the put price decreases

34 What about Variance? When an option may finish out-of-the-money (expire without being exercised), there is another factor that helps determine price The variance in underlying asset returns is a less obvious, but important, determinant of option values The greater the variance, the more the call and the put are worth If an option finishes out-of-the-money, the most you can lose is your premium, no matter how far out it is The more an option is in-the-money, the greater the gain The owner of the option gains from volatility on the upside, but don’t lose anymore from volatility on the downside

35 Table 14.2

36 Two-period bopm example

37 Two-period bopm example

38 Put option and put-call parity

39 Put option and put-call parity

40 Protective Put Buy the underlying asset and a put option to protect against a decline in the value of the underlying asset Pay the put premium to limit the downside risk Similar to paying an insurance premium to protect against potential loss Trade-off between the amount of protection and the price that you pay for the option

41 An Alternative Strategy You could buy a call option and invest the present value of the exercise price in a risk-free asset If the value of the asset increases, you can buy it using the call option and your investment If the value of the asset decreases, you let your option expire and you still have your investment in the risk- free asset

42 Comparing the Strategies Stock + Put If S < E, exercise put and receive E If S ≥ E, let put expire and have S Call + PV(E) PV(E) will be worth E at expiration of the option If S < E, let call expire and have investment, E If S ≥ E, exercise call using the investment and have S Hence, we have the put-call parity Value at Expiration Initial PositionS < ES ≥ E Stock + PutES Call + PV(E)ES

43 Employee Stock Options Options that are given to employees as part of their benefits package Often used as a bonus or incentive Designed to align employee interests with stockholder interests and reduce agency problems Empirical evidence suggests that they don’t work as well as anticipated due to the lack of diversification introduced into the employees’ portfolios The stock isn’t worth as much to the employee as it is to an outside investor because of the lack of diversification – this suggests that options may work in limited amounts, but not as a large part of the compensation package Esos cannot be sold, have much longer life, vesting period, Occasionally underwater Esos are repriced

44 Equity: A Call Option Equity can be viewed as a call option on the company’s assets when the firm is leveraged The exercise price is the face value of the debt If the assets are worth more than the debt when it comes due, the option will be exercised and the stockholders retain ownership If the assets are worth less than the debt, the stockholders will let the option expire and the assets will belong to the bondholders

45 Example Consider a firm Single debt issue, face value $1,000, zero coupon bond and due in one year Current market value of firm’s assets is $980 Risk free rate is 12.5% Equity is a call option with exercise price being equal to the face value of debt

46 Case 1:Debt is risk free In one year, the value of firm’s assets will be either $1,100 or $1,200. What is the current market value of debt? What is the ytm?

47 Case 2:Debt is risky In one year, the value of firm’s assets will be either $800 or $1,200. Current value of the debt=$980-$134.44=$845.56

48 Warrants A call option issued by corporations in conjunction with other securities to reduce the yield required on the other securities Differences between warrants and traditional call options Warrants are generally very long term They are written by the company and warrant exercise results in additional shares outstanding The exercise price is paid to the company, generates cash for the firm, and alters the capital structure Warrants can normally be detached from the original securities and sold separately Exercise of warrants reduces EPS, so warrants are included when a firm reports “diluted EPS”

49 Convertibles Convertible bonds (or preferred stock) may be converted into a specified number of common shares at the option of the bondholder The conversion price is the effective price paid for the stock The conversion ratio is the number of shares received when the bond is converted Convertible bonds will be worth the straight bond value or the conversion value, whichever is greater

50 Valuing Convertibles Suppose you have a 10% bond that pays semiannual coupons and will mature in 15 years. The face value is $1,000 and the yield to maturity on similar bonds is 9%. The bond is also convertible with a conversion price of $100. The stock is currently selling for $110. What is the minimum price of the bond? Straight bond value = 1, Conversion ratio = 1,000/100 = 10 Conversion value = 10*110 = 1,100 Minimum price = $1,100

51 Valuing Convertibles The value of a convertible bond will always exceed the floor value unless the firm is in default or bondholders are forced to convert.

52 Other Options Call provision on a bond Allows the company to repurchase the bond prior to maturity at a specified price that is generally higher than the face value Increases the required yield on the bond – this is effectively how the company pays for the option Put bond Allows the bondholder to require the company to repurchase the bond prior to maturity at a fixed price Insurance and Loan Guarantees These are essentially put options Fire insurance: In case of fire, you exercise your put option and force the insurer to pay you the amount of insurance. If you lend money to someone and they default, then, with a guaranteed loan, you can collect from someone else, often government.